MATH 111 - Sec 2.2-2.8: Equations & such
sinusoidal graphs: what are the 5 key points on each graph: y = sinx , on the interval 0 ≤ x ≤ 2π
(0, 0), (π/2, 1), (π, 0), ((3π)/2, -1), (2π, 0)
sinusoidal graphs: what are the 5 key points on each graph: y = cosx , on the interval 0 ≤ x ≤ 2π
(0, 1), (π/2, 0), (π, -1), ((3π)/2, 0), (2π, 1)
steps to finding an equation for a sinusoidal graph
1. Determine which function the graph has characteristics of (if the graph starts on the x-axis, it is a sine function) 2. determine the period and amplitude 3. from the period, determine the value of w using the equation T = (2π)/w 4. write equation in form y = A sin (wx) or y = A cos (wx)
reciprocal identities (3)
1. cscθ = 1/sinθ 2. secθ = 1/cosθ 3. cotθ = 1/tanθ
steps to graphing a sinusoidal function of the form y = A sin (wx) or y = A cos (wx)
1. determine the amplitude ( |A| ) 2. determine the period using w and T = (2π)/w 3. divide the period ((2π)/w) into 4 subintervals of equal lengths 4. use the end points of these intervals to obtain 5 key points on the graph 5. plot the 5 key points w/ a sinusoidal graph to obtain the graph of one cycle 6. extend the graph in each direction to make it complete
steps to finding the exact value of a trig function using a reference angle?
1. draw the original angle 2. find the reference angle = the acute angle made by the x-axis and the terminal side of the original angle 3. then, find the exact value of the reference angle 4. adjust the (+ or -) sign of the value of the trig function, depending on which quadrant the angle lies in.
steps to finding the exact value of sin, cos, csc, or sec function using periodic properties?
1. if θ is in degrees: θº - 360º = smaller angle in degrees; find the exact value of the smaller angle 2. if θ is in radians: θπ - 2π = smaller angle in radians; find the exact value of the smaller angle
Quotient Identities (2)
1. tanθ = sinθ/cosθ 2. cotθ = cosθ/sinθ
Domain and Range of the Trig Function: cosine
>domain: all real numbers (-∞ < cosθ < ∞) >range: -1 ≤ cosθ ≤ 1 (all real numbers from -1 to 1)
Domain and Range of the Trig Function: sine
>domain: all real numbers (-∞ < sinθ < ∞) >range: -1 ≤ sinθ ≤ 1 (all real numbers from -1 to 1)
Domain and Range of the Trig Function: cotangent
>domain: all real numbers, except multiples of π (180º) >range: all real numbers (-∞ < cotθ < ∞)
Domain and Range of the Trig Function: cosecant
>domain: all real numbers, except multiples of π (180º) >range: cscθ ≤ -1 OR cscθ ≥ 1 (all real numbers greater than or equal to 1 or less than or equal to -1)
Domain and Range of the Trig Function: tangent
>domain: all real numbers, except odd integer multiples of π/2 (90º) >range: all real numbers (-∞ < tanθ < ∞)
Domain and Range of the Trig Function: secant
>domain: all real numbers, except odd integer multiples of π/2 (90º) >range: secθ ≤ -1 OR secθ ≥ 1 (all real numbers greater than or equal to 1 or less than or equal to -1)
area of a sector equation = ?; units = ?
A = ½ r² θ (r = radius); length²
sinusoidal graphs: if w<0, we use even-odd properties; what are these two even-odd property equations for sin and cos?
A sin (-wx) = -A sin (wx) A cos (-wx) = A cos (wx)
state whether each trig function is positive or negative in the following quadrant: I
all six trig functions are positive
complementary angle theorems: cos B = ?
cos B = a/c = sin A
complementary angle theorems: cos θ = ?
cos θ = sin (90º - θ) cos θ = sin (π/2 - θ)
Even-Odd Properties: give equation, tell whether even or odd and symmetric with respect to what? cos
cos(-θ) = cosθ even symmetric to y-axis
complementary angle theorems: cot B = ?
cot B = a/b = tan A
complementary angle theorems: cot θ = ?
cot θ = tan (90º - θ) cot θ = tan (π/2 - θ)
Even-Odd Properties: give equation, tell whether even or odd and symmetric with respect to what? cot
cot(-θ) = -cotθ odd symmetric to origin
complementary angle theorems: csc B = ?
csc B = c/b = sec A
complementary angle theorems: csc θ = ?
csc θ = sec (90º - θ) csc θ = sec (π/2 - θ)
Even-Odd Properties: give equation, tell whether even or odd and symmetric with respect to what? csc
csc(-θ) = -cscθ odd symmetric to origin
what equation do you use first when finding the exact values of the trig functions using a point (a,b) on the angleθ's terminal side?
r = √(a²+b²)
arc length equation = ?; units = ?
s = rθ (s = arc length, r = radius, θ = angle); length
complementary angle theorems: sec B = ?
sec B = c/a = csc A
complementary angle theorems: sec θ = ?
sec θ = csc (90º - θ) sec θ = csc (π/2 - θ)
Even-Odd Properties: give equation, tell whether even or odd and symmetric with respect to what? sec
sec(-θ) = secθ even symmetric to y-axis
complementary angle theorems: sin B = ?
sin B = b/c = cos A
what are the equations for finding the exact values of each of the trig functions if P = (a,b) is the point on the unit circle that corresponds to the real number t ?
sin t = b cos t = a tan t = b/a (a≠0) csc t = 1/b (b≠0) sec t = 1/a (a≠0) cot t = a/b (b≠0)
sinusoidal graphs: sin x = cos ?
sin x = cos (x - π/2)
using the equation, r = √(a² +b²); what are the ratios of the six trig functions of θ?
sin θ = b/r cos θ = a/r tan θ = b/a csc θ = r/b sec θ = r/a cot θ = a/b
complementary angle theorem: sin θ = ?
sin θ = cos (90º - θ) sin θ = cos (π/2 - θ)
Even-Odd Properties: give equation, tell whether even or odd, and symmetric with respect to what? sin
sin(-θ) = -sinθ odd symmetric to origin
what are the equations for finding the period of each trig function graph?
sin, cos, csc, sec: T = (2π)/w tan, cot: T = π/w
Pythagorean Identities (3)
sin²θ + cos²θ = 1 tan²θ + 1 = sec²θ cot²θ + 1 = csc²θ
state whether each trig function is positive or negative in the following quadrant: III
sinθ, cscθ = negative cosθ, secθ = negative tanθ, cotθ = positive
state whether each trig function is positive or negative in the following quadrant: IV
sinθ, cscθ = negative cosθ, secθ = positive tanθ, cotθ = negative
state whether each trig function is positive or negative in the following quadrant: II
sinθ, cscθ = positive cosθ, secθ = negative tanθ, cotθ = negative
complementary angle theorems: tan B = ?
tan B = b/a = cot A
complementary angle theorems: tan θ = ?
tan θ = cot (90º - θ) tan θ = cot (π/2 - θ)
Even-Odd Properties: give equation, tell whether even or odd and symmetric with respect to what? tan
tan(-θ) = -tanθ odd symmetric to origin
convert degrees to radians
θº × (π/180)
convert radians to degrees
θπ × (180/π)